This problem is part of a class of problems known as ternary Diophantine equations or generalized Fermat equations: one asks whether

A^p + B^q = C^r (**)

has any solution in relatively prime integers (A,B,C).

(Exercise: show that if we don’t impose the condition that (A,B,C) be relatively prime, there are always infinitely many solutions to (**), so it’s not such an interesting question.)

We denote by GF(p,q,r) the problem of finding all solutions to (**). The most famous case, of course, is that where p=q=r! This is the classical conjecture of Fermat, finally resolved by Wiles and Taylor-Wiles in 1995 after a few hundred years of hard work by many hands.

You might ask: what if it’s not possible to write down all solutions to GF(p,q,r) — what if, for instance, there are infinitely many? A beautiful theorem of Henri Darmon and Andrew Granville demonstrates that, apart from very small values of p,q,r, this isn’t possible.

The proof reduces the problem to the famous theorem of Faltings, that a curve of genus at least 2 has only finitely many rational points. But there’s actually a very nice heuristic argument which (disadvantage) doesn’t prove anything, but (advantage) makes it clear why this funny condition 1/p + 1/q + 1/r < 2 is relevant.

Heuristic: In the interval [0..X] there are about X^(1/p) pth powers, and about X^(1/q) qth powers. So the number of sums of the form A^p + B^q is about X^(1/p + 1/q). Now what’s the chance that you get incredibly lucky and this sum is actually a perfect rth power? Well, the sum is also going to be in [0..X] (more precisely [0..2X], but we won’t trouble ourselves with constants!) There are X^(1/r) rth powers in this interval, so the chance that any particular number is an rth power is X^(1-1/r). So the expected number of solutions to (**) in this interval is

X^(1/p + 1/q + 1/r – 1)

and what the Darmon-Granville condition says is precisely that the exponent is negative — that in a large interval we should expect to see approximately zero solutions.

In general, very little is known about GF(p,q,r). A very incomplete sampling of existing results: the cases of GF(p,p,2) and GF(p,p,3) were handled by Darmon and Loic Merel in 1997; (3,3,p) by Alain Kraus in 1998; (2,3,8) and (2,3,9) by Nils Bruin in 2000 and 2004; and (2,3,7), in a technical tour de force by Bjorn Poonen, Ed Schaefer, and Michael Stoll, just last year. The only solution to this last equation (in positive integers) is

15312283^2 + 9262^3 = 113^7.

Now things get a bit more technical. A few years ago, I proved that GF(2,4,p) had no solutions for p > 211. The proof follows the general lines of the Wiles argument, using theorems about modularity of elliptic curves. However, I had to use a step from analytic number theory — namely, I needed to estimate the average special value of a family of L-functions, using an idea of Bill Duke. The fact that I could only estimate this average, not compute it exactly, led to the fact that I could only handle GF(2,4,p) for fairly large values of p. So a few times a year people would ask me, “What about smaller values of p? Can they be handled?” Now I can say yes — between us, we were able to bring to bear enough techniques to settle the problem completely. What goes into it: a sharpening of my original estimate, some applications of Chabauty techniques a la Bruin, elimination of various p by arguments on Galois representations, and finally direct computation of special values in MAGMA to take care of the remaining cases. The result is a bit of a patchwork, and I don’t know if it’ll be fun to read — but it was fun to do!